1. Field of the Art
The present invention consists of a system and method for obtaining Tomographic Images of the Primary Electric Currents (PEC) produced by the neurons of the brain and the muscle cells of the heart.
2. Description of the Prior Art
In this invention reference will be made to phrases and terms used of art that will be defined as follows:
Vectors will usually be denoted with boldface lowercase letters and matrices usually with uppercase boldface letters. 1N is the vector of dimension N of ones. In denotes the identity matrix of order n.  is the matrix Kronecker product. “.*” is the Hadamaard product and o represents the dyadic product among vectors. If x is a matrix or a vector, then xt denotes x transposed x; x* denotes x conjugate transposed. The operation vech(X) applied to X consists of forming a vector with the columns of X, eliminating repeated elements or those considered constant in the context of the problem.
Primary Electric Current (PEC) jP(r,t): is the macroscopic magnitude obtained by the spatial and temporal average of the post-synaptic activity of a group of nervous or heart cells at the position {right arrow over (r)} and the instant of time t.
Volume Conductor ΩV: the region inside the body that will be the object of study (e.g., the head and/or the torso.
Generating Volume Ωg: the subset of ΩV where the PEC originates (i.e., the brain or the heart).
Lattice of the Volume Conductor V: the discrete group of NV points rvεΩV.
Lattice of the Generating Volume g: the discrete group of Ng points rgεΩg.
Lattices of Time Instants ℑm(tI,I=1, . . . ,NIm): time scale for which each selected functional image m has been sampled.
Analysis Interval ℑ=[0,T] is the period defined from a reference fixed arbitrarily at the instant 0 until time T.
PEC Tomographic Image (TPEC): vector j(t)=└jP({right arrow over (r)}g,t)┘1≦g≦Ng defined at the points rgεΩg, and for each instant tεℑ of time.
Electro-encephalogram (EEG) and Electro-cardiogram (EKG): time series obtained by measuring, on the head and the thorax respectively, the voltage difference Ver(t) that is created in a pair of electrodes localized on the body in the positions re (recording electrode) and rr (reference). Ver(t) is measured at Ne sites on the body. A vector of measurements of this type will be denoted by v(t).
Magneto-encephalogram (MEG) and Magneto-cardiogram (MKG): time series obtained by measuring the projection bcn(t) of the magnetic field density vector at the center rc of a simple coil on nc, the perpendicular vector to the plane that contains it. These simple coils are connected to Superconducting Quantum Interference Devices (dc SQUID) by means of a group of magnetic flow detection transformers. bcn(t) are measured at Nc sites of the body. The vector of measurements of this type will be denoted by b(t).
Anatomical Images: type of medical images that offer structural information about the body such as Computed Axial Tomography (CAT), Magnetic Resonance Images (MRI), post-mortem sections of the head using cryotomy, etc.
Anatomical Atlas: is an anatomical image of the brain or of the heart in a reference system of the head or the torso, respectively. A particular instance of a reference system for the brain is the “Talairach” international system. Possible types of anatomical atlases are:                Individual: structural images of the subject under study (FIG. 1); and        Probabilistic: a composite statistical image that summarizes the inter-individual variability of normal or pathological morphologies of anatomical images in a given population (Valdés P. and Biscay R. The statistical analysis of brain images. In: Machinery of the Mind, E. Roy John et al. the. (ed.), (1990). Birkhauser, pp. 405–434.; Collins D L, Neelin P, Peter T M, Evans A C (1994) Automatic 3D registration of MR volumetric dates in standardized talairach space. J Comput. Assist. Tomogr. 18(2): 192–205; Evans A C, Collins D L, Mills S R, Brown E D, Kelly R L, Peters T M (1993) 3D statistical neuroanatomical models from 305 MRI volumes. Proc. IEEE-nuclear Science Symposium and Medical Imaging Conference: 1813–1817; Evans, A. C., Collins, D. L., Neelin, P., MacDonald, D., Kambei, M., and Marret, T. S. (1994) Three dimensional correlative imaging: Applications in human brain mapping. In R. Thatcher, M. Hallet, T. Zeffiro, E. Roy John and M. Huerta (Eds.) Functional neuroimaging technological foundations. (Academic Press).        
Functional Images: types of medical images that offer information on the hemodynamics corporal metabolism such as: Functional Magnetic Resonance (fMRI), Positron Emission Tomography (PET), and the Single Photon Emission Tomography (SPECT).
Observations o(t)=[o(t)m]1≦m≦M the group of M measurements of different modalities (EEG, MEG, and functional image) that will be used for the construction of the TPEC. All the measurements are referred to the reference system of the selected anatomical atlas. The EEG/EKG (o1(t)) and/or the MEG/MKG (o2(t)) and optionally the fMRI, PET, and SPECT (om(t), m>2) will be always included in o(t). These observations are defined for the lattices of sensor positions m as the set of Cartesian coordinates that define the position and orientation of each sensor in a predefined common body reference system, said lattices being designated 1, 2, 3, 4, and 5 for the EEG/EKG, MEG/MKG, fMRI, PET, and SPECT, respectively. Alternatively, o(t) may be defined for the coordinates yεΔm, where Δm is the Space of Measurement of the modality m.
Multivariate Z Transform of the vector of random variables xΘx={μx, Σx}, their mean vector and matrix of population covariances is defined
      z    X    =            (              x        -                  μ          x                    )        ·                            ∑          x                          -                      1            2                              .      In this last expression one of the square roots of Σx is selected and, if this is not of complete range, a pseudoinverse is used. In the case that x is of dimension 1, its Z Transform has the simple expression
      z    X    =                    x        -                  μ          x                            σ        x              .  
Generalized Gaussian Distribution Ntk,p(μx,Σx): is a probability density proportional to exp(−β∥x−μx∥Σxp), where k is the dimension of the distribution, p the order of the norm in the exponent ∥x−μx∥Σxp=|zx|p, β a proportionality constant, and t is the numeric variable type (i.e., R—real or C—complex). For example, the notation x˜NR2,p(μx,Σx) is used to express that x is a sample from a multivariate real gaussian distribution.
Functional Space: is a set of functions with common properties (Triebel, H. 1990. Theory of Function Spaces II. Basel: Birkhauser). Membership in a function space is used to specify desirable properties in the TPEC under consideration. The function spaces are considered to be consisting of the combination of atoms.
Dictionary of Atoms: is a collection of atoms with properties such that all element of a function space can be expressed as
      f    =                  ∑        k            ⁢                        F          k                ·                  ψ          k                      ,where ψk are the atoms. Examples are: “Wavelets”, (Meyer, Y. 1992. Wavelets and Operators. Cambridge: Cambridge University Press), the Fourier base of complex exponential, the collection of Dirac deltas in a space of generalized functions, etc. A Megadictionary is the union of several dictionaries of atoms.
Besov Space Bn,sm(Ω→Rn): is a collection of functions defined on the space Ω and taking values in Rn that possess a degree of smoothness specified by means of the triple (m, n, s) (Triebel, H. 1990. Theory of Function Spaces II. Basel: Birkhauser). The requirement that any given function f belongs to Bn,sm shall be imposed by means of penalization with the metric
                                        f                                    B                      n            ,            s                    m                    ≈                                  f                                    b                      n            ,            s                    m                      =                  ∑        k            ⁢                        a          k                ·                                                        F              k                                            m                      ,that is to say the norm of weighted combination of the atoms (De Vore R. A. and Popov V., (1988) Interpolation of Besov Spaces, Transactions of the American Mathematical Society 305, 397–414). These spaces allow the modeling of functions with controllable degree of spatial inhomogeneity. Particular cases are:                Sobolev Space: B2,2m=Hm, of maximum smoothness (with respect to derivatives of order m); and        The “algebra of bumps”: B1,s1 that allows modeling both of point sources (dipoles) and distributed sources in a common framework. (Meyer Y., (1992) Wavelets and Operators, Cambridge: Cambridge University Press).        
Reference group: consists of a reference group, the membership to which is to be established for a given TPEC image.
Classical Quantitative Electrophysiology (qEEG/qEKG)
This invention has as antecedents the work directed to quantify and to make objective the detection of abnormal states of the brain and heart by applying multivariate statistical analysis to the EEG and EKG. These methods are known as quantitative electroencephalography and quantitative electrocardiography (abbreviated respectively as qEEG and qEKG). Systems and methods for qEEG were described in the patents U.S. Pat. No. 4,846,190; U.S. Pat. No. 4,913,160; U.S. Pat. No. 5,282,474 and U.S. Pat. No. 5,083,571. Methods for qEKG were described in the U.S. Pat. No. 4,974,598. These patents specify recording of the EEG and/or EKG (v(t)) by means of a plurality of sensors.
Optionally, instead of analyzing the v(t) recorded originally, pre-processed series (PPS) are obtained by means of the calculation of the cross-covariance function of v(t) with respect to a time series s(t), marker of external events. In the case of the EEG, s(t) may be taken as a sequence of Dirac delta functions indicative of the moments at which certain stimuli are presented to the subject examined. The resulting time series is known as the Average Evoked Potential (AEP). In the case of the EKG, s(t) may signal the occurrence of the R wave of the EKG itself and the resulting PPS is the average EKG (AEKG).
The extraction of Descriptive Parameters (DP) from these time series are statistical summarizations of the recorded time series designed to reflect variations in normal and pathological physiological activity.
In the above-mentioned patents, the DP specified for the EEG are the averages in broad bands of the frequency of spectrum, and elementary transformations of these DP are also included. The DP of the Broad Band Spectral Analysis (BBSA) are defined formally as follows: SO(ω) is the crosspectral or variance and covariances matrices of the coefficients of Fourier of V(t) (where ω denotes frequency). The DP-BBSA are:
            ∫      a      b        ⁢                            s                      O            ,            ij                          ⁢                  (          ω          )                    ⁢              ⅆ        ω              ,where a and b specify the limits of the broad band and sO,ij(ω) is the element i, j of the matrix SO(ω).
In the patents cited, the DP for the AEP and AEKG are defined as the coefficients of a Karhunen-Loeve bases. These parameters are known as Factorial Analysis loadings.
In the patents cited, transformations of the DP are carried out to guarantee their gaussianity. In the description of this invention a generic transformed scalar DP will be denoted with the letter x=T(DP) and a vector with the notation x=T(DP), after the transformation x˜NR2,p(μx,Σx).
In the patents cited, a comparison of the DP is carried out with respect to the normal variability (inferred from normative databases) by means of the univariate z transform that expresses the DP as a deviation measured in units of standard deviation σx, with respect to the populational average μx. In the calculation of the zx based on the normative database, the effect of concomitant variables that originate variability that is not of diagnostic interest (as is the case of the subject's age), is eliminated by means of homoscedastic polynomial regression.
In the construction of brain and heart Topographic Maps (TM), the degree of deviation of the subject with respect to the norm is coded by means of a color scale. The TM is an image interpolated between the measurements of the sensors, which represents a schematic two-dimensional projection of the head or of the torso.
Cluster analysis is used to define groups of similar subjects according to the values of their DP (see U.S. Pat. No. 5,083,571).
Linear discriminant analysis is used to classify a subject examined as belonging to a test group or a test group previously defined by cluster analysis on the basis of predefined values of their DP (see U.S. Pat. No. 5,083,571).
The measurement of the statistical distances of a subject with respect to his own previous states is carried out at selected moments. This offers information on the subject's physiological state, for example, in the following conditions: during an operation, while in intensive therapy, or while in evaluation for the evolution of pathology (U.S. Pat. Nos. 4,545,388; 4,447,270; 4,815,474; 4,844,086 and 4,841, 983).
The utility of these methods has been confirmed in several studies (John, E. R.; Harmony, T. Valdés-Sosa and, P. (1987b): The uses of statistics in electrophysiology. In: Gevins, A. S. and Rémond, A. (Eds), Handbook of Electroencephalography and Clinical Neurophisiology. Revised Series. Volume 1, Elsevier, The Netherlands, 497–540 and John, E. R.; Prichep, L. S. and Easton, P. (1987a): Normative data bases and neurometrics. Basic concepts, methods and results of norm construction. In: Gevins, A. S. and Rémond, A. (Eds), Handbook of Electroencephalography and Clinical Neurophisiology. Revised Series. Volume 1, Elsevier, The Netherlands, 449–496).
However, these methods have the following limitations:
The DP-BBSA are insufficient as descriptive parameters for the quantification of many types of activities of the EEG. This is due to the loss of resolution caused by the averaging over bands in frequency, with the resulting loss of sensitivity and specificity in diagnoses (Szava, S.; Valdés, P.; Biscay, R.; Galán, L.; Bosch, J.; Clark, I. and Jiménez, J. C.: High Resolution Quantitative EEG Analysis. Brain Topography, Vol. 6, Nr. 3, 1994, pp. 211–219).
The analysis of a group of DP by use of TM is hindered by the high correlation among the variables. To overcome this difficulty, Galán et al. (Galán, L.; Biscay, R.; Valdés, P.; Neira, L. and Virués T. (1994): Multivariate Statistical Brain Electromagnetic Mapping. Brain Topography, vol 7. No.1) introduced the Multivariate TM by means of the use of the multivariate Z transform.
The evaluation of the normality of the TM is carried out by means of univariate statistics without taking into account the need for the control of Type I errors caused by the multiple comparisons for all point of the map. This increases to uncontrollable levels the probability of false anomaly detection.
The procedure for the statistical classification of subjects is based on the assumption that x˜NR2,p(μx,Σx). This implies that the decision frontiers between the groups are linear which is too restrictive. In addition, optimal procedures for the selection of the DPs with highest classification power have not been applied.
The use of magnetic measurements b(t) are not included, in spite of the fact that these add additional information to that provided by v(t).
The analysis of physiological information is limited to v(t), without there being any inference about j(t). Therefore, none of these methods constitutes a modality of TPEC. They are merely an analysis of projections of j(t) on the surface of the body after being distorted by the tissues of the subject interposed between the generators of the neural/cardiac PEC and the sensors. In fact, the relationship between the PEC and the EEG/MEG/EKG/MKG depends on the conductor properties of the head and the torso (e.g. geometry, conductivity, electric and magnetic permeability, etc.).
Cerebral and Heart Multivariate Statistical Maps
Some of these limitations were overcome by the U.S. Pat. No. 5,282,474 of Pedro Valdés-Sosa et al. In this patent the use of statistical methods is claimed for the evaluation of the abnormal activity of the brain and of the heart with the following innovations:
(1) b(t) is added as an optional source of physiological information.
(2) Additional components of the time series s(t) are included in the form of markers of external events of more general type such as can be derived from the subject's own voluntary or involuntary reactions, or those obtained by the analysis of their own o(t). The series derived by such a pre-processing are the Event Related Components (ERC) derived from o(t).
(3) Expansion of the ERC in tensor products of functional space-time bases. These include not only the basis of Fourier, but also wavelets (for the description of non-stationary processes) among other sets of DP that increase flexibility in the description of physiological processes.
(4) Summarize inclusive of these DP by means of the use of higher order statistical moments and different parametric time series models. In particular the coefficients of linear autorregresive are included.
(5) The introduction as DP of High Resolution Spectral Analysis (HRSA) consistent in the use of SO(ω) for all the frequencies ω according to the one transformed of discrete Fourier; and
(6) The introduction in the TM of color scales based on the empirical probability distribution of the global maxima and global minima of the DP. This scale exercises an effective control of the probability of false pathology detection.
The improvements for the evaluation of brain activity introduced in the Sosa et al. '474 patent were exemplified as follows:
(1) By means of the construction of norms of HRSA for the Cuban population from 5 to 97 years (Valdés, P.; Biscay, R.; Galán, L.; Bosch, J.; Szava, S.; and Virues, T.: High Resolution Spectral EEG norms for topography. Brain Topography, 1990, vol. 3, pp. 281–282 and Valdés, P.; Bosch, J.; Serious of it Banks, R.; Hernández, J. L.; Pascual, R. and Biscay, R.: Frequency domain models for the EEG. Brain Topography, 1992a, vol. 4, pp. 309–319); and
(2) By the showing that the DP-HRSA achieve a higher sensitivity and specificity than the DP-BBSA for the detection of neurological and psychiatric pathology (Szava, S.; Valdés, P.; Biscay, R.; Galán, L.; Bosch, J.; Clark, I. and Jiménez, J. C.: High Resolution Quantitative EEG Analysis. Brain Topography, Vol. 6, Nr. 3, 1994, pp. 211–219).
However, the construction of DP of o(t) used in the Sosa et al. '474 patent is limited to such parametric models as the DP-HRSA. These are only a complete description of o(t) for stationary and linear stochastic signals or those signals with limited types of nonlinearity. The inadequacy of such DP for the study of the EEG of some types of patient (i.e., those with epilepsy) was demonstrated by Hernández, Valdés-Sosa and Vila (Hernández, J, L., Valdés, P. A., and Vila, P. (1996) EEG spike and wave modeled by to stochastic limit cycle. NeuroReport, 7: 2246–2250).
Moreover, qEEG methods and qEKG are not applied to estimates of j(t). For this reason, they are not true variants of TPEC.
Tomography of Cerebral and Heart Primary Electric Currents
As in all type of Tomography, the starting point for the development of the TPEC is the establishment of a model that relates the observed data with the quantity to estimate, this model is known as the Direct Problem. In the TPEC, the direct problem postulates how the o(t) are generated the from j(t). This model has two components:
(1) The specific model of Volume Conductor assumed, that is to say, of the conductive properties of the head and the torso, in particular their geometry, conductivity, electric and magnetic permeability, etc; and
(2) The model that is assumed for j(t), or source model; and
The properties of the Volume Conductor are summarized in the electric kE(r) and magnetic kM(r) Lead Fields (LF). These are the kernels of Fredholm integral equations of the first kind that establish direct relationships between the PEC jP(r,t) with Ver(t) and bcn(t).
                                          V            er                    ⁡                      (            t            )                          =                              ∫                          Ω              v                                ⁢                                                                      k                  E                                ⁡                                  (                  r                  )                                            ·                                                j                  P                                ⁡                                  (                                      r                    ,                    t                                    )                                                      ⁢                          ⅆ                              r                3                                                                        (        1        )                                                      b            cn                    ⁡                      (            t            )                          =                              ∫                          Ω              v                                ⁢                                                                      k                  M                                ⁡                                  (                  r                  )                                            ·                                                j                  P                                ⁡                                  (                                      r                    ,                    t                                    )                                                      ⁢                          ⅆ                              r                3                                                                        (        2        )            
The discretization of the terms expressed in equations (1) and (2) lead to the following formulation of the direct problem:o(t)=K·j(t)+e(t)  (3)where K is the discretized LF and e(t)˜.Ntk,p(0,ΣEE), being the error introduced at the sensors in the moment of the measurement.
The use of the LFs to formulate the Direct Problem instead of using the classic Green formulation results in remarkable advantages according to Rush and Driscoll (Rush S. and Driscoll D. A. EEG electrode sensitivity—an application of reciprocity. IEEE Trans. on Biomed. Eng., vol. BME-16, pp. 15–22, 1969) and Plonsey (Plonsey R. Capability and limitations of electrocardiography and magnetocardiography. IEEE Trans. Biomed. Eng., vol. BME-19, not. 3, pp. 239–244, 1972). This is mainly because all the conductive properties of the head or the thorax can be summarized in these magnitudes without assuming a specific model for the PEC.
However, it is well-known that the conductivity in the body is non homogeneous and non isotropic (Hoeltzell P. B. and Dykes R. W. Conductivity in the somatosensory cortex of the cat. Evidence for cortical anisotropy. Brain Research, 117, pp. 61–82, 1979), a fact that complicates the calculation of the LFs considerably. Nevertheless, a substantial simplification is possible when the conductivity is considered constant and isotropic in each tissue type, which constitute a compartment (Schwan H. P. and Kay C. F., The conductivity of living tissues. Ann. N.Y. Acad. Sci., vol. 65, pp. 1007–1013, 1957). This model of Volume Conductor is denominated Isotropic and Piecewise Homogeneous Volume Conductor (IPHC). In general, two types of model IPHC are in use for the head or the torso:
(1) Spherical: the simplest model that specifies that the surfaces defining the different compartments are concentric spheres. For this model explicit formula exist to evaluate K. This model is easy to evaluate but not very exact, mainly for the temporal regions of the head; and
(2) Realistic: in which the surfaces of the compartments are obtained from an anatomical atlas. For this model numeric methods have been developed by Fletcher et al. (Fletcher D. J., Amir A., Jewett D. L. and Fein G. Improve method for the calculation of potentials in a realistic head shape model. IEEE Trans. Biomed. Eng., vol 42, not. 11, pp. 1094–1104, 1995) and Oostendorp and van Oosteroom (Oostendorp T. and van Oosterom A. The potential distribution generated by surface electrodes in homogeneous volume conductive of arbitrary shape. IEEE Trans. Biomed. Eng., vol. BME-38, pp. 409–417, 1991). However, this procedures is based on a boundary element method (BEM) for the calculation of the potential difference on the surfaces that limit the compartments produced by energizing the sensors.
The expression:RO=∥o(t)−K·j(t)∥ΣEEpO  (4)is the functional of the Direct Problem in the norm pO. l(o(t)|(t),ΣEE)=CO exp[−Ro] is defined as the likelihood corresponding to equation (3), being C0 the constant of normalization of the density. In what follows, unless otherwise stated, pO=2.
The central objective of the TPEC is the estimation of j(t) by means of the minimization of RO=∥o(t)−K·j(t)∥ΣEEpO, which is the “Inverse Problem”. It is well known that RO=∥o(t)−K·j(t)∥ΣEEpO lacks, in general, a unique solution since the LF operator has a non-trivial null space, even when simultaneous electric and magnetic measurements are available.
Estimation of the PEC is possible only if a priori information is contributed in the form of a model for j(t) that impose a series of restrictions. These may be classified as:
Extrinsic: consisting of requiring the compatibility of the desired solution with information provided from other types of neuroimages. About the cerebral anatomical neuroimages, typical requirements are: (1) that j(t) be estimated only inside the spherical Volume Conductor that models the head; and (2) that j(t) be restricted to the cortical surface of the of the subject with an orientation perpendicular to said cortex (Dale, A. M., and Sereno and, M. I., J. Cognit. Neurosc., 1993, 5:2, pp. 162–176).
Intrinsic: consisting of requirements about the shape and extent of areas of activated tissue. It is required that the activated areas are as small as possible while maintaining compatibility of the model with the data (simplicity). As for the variation in the shape in the space, j(t), this requirement has ranged from minimal smoothness (collections of Dirac deltas corresponding to current dipoles) to functions that are maximally smooth.
All the above stated conditions are imposed on the solution in two equivalent ways:
Regularization: Adding to RO=∥o(t)−K·j(t)∥ΣEEpO a non-negative functional RJ=RJ(j(t)|Θ) and minimizing RO+λ·RJ(j(t)|Θ).
Bayesian estimation: Maximizing the a posteriori distributionP(j(t),ΣEE|o(t))∝l(o(t)|j(t),ΣEE)·π(j(t)|Θ)  (6)where π(j(t)|Θ)=CJ exp[−RJ] is the a priori probability of j(t).
The fundamental differences in the different methods that have been developed to estimate the generators of o(t) are in the model specified for j(t). These are reviewed as follows:
When RJ(j(t)|Θ) is a functional that guarantees that Ng<Θo, where Θo is a constant that assures the uniqueness of the solution of RO=∥o(t)−K·j(t)∥ΣEEpO. This class of models for j(t) is denominated current dipoles (Scherg, M. and Ebersole, J. S. (1993): Models of Brain Sources. Brain Topography. Vol. 5, Nr. 4, pp. 419–423).
                              j          ⁡                      (                          r              ,              t                        )                          =                              ∑                          g              =              1                                      N              g                                ⁢                                                                      α                  g                                ⁡                                  (                  t                  )                                            ·                                                μ                  g                                ⁡                                  (                  t                  )                                                      ⁢                          δ              ⁡                              (                                  r                  -                                      r                    g                                                  )                                                                        (        7        )            
Where αg(t) is the activity of the generator g and μg(t) its orientation. A model of this type was described in the U.S. Pat. Nos. 4,949,725; 5,285,385; 5,524,086, and 5,361,774. The advantages of these models are: (1) they are effective for modeling situations in which a small group of regions is producing PEC, all of small extent, (2) the formulation leads to simple least squares estimation methods, (3) the set of resulting DP of the estimates is very compact, comprising the positions and orientations of each current dipole, and (4) due to the simplicity of the model, it is easy to include additional restrictions to the estimation of this model of PEC (Scherg, M. and Ebersole, J. S. (1993): Models of Brain Sources. Brain Topography. Vol. 5, Nr. 4, pp. 419–423). For example a certain degree of smoothness may be imposed on αg(t) and the requirement that, μg(t)=μg does not depend on time. These requirements stabilize the estimated DP.
Nevertheless, the dipolar models discussed have the following limitations:
(1) When the number of possible generators of PEC (Ng) becomes too large, this type of model requires of an excessive manual intervention of the operator. For this case, the inverse problem becomes again ill determined. In fact, these models do not include a statistical approach for determining Ng.
(2) When the different regions of tissue that generate PEC are distributed in extensive areas, the dipolar model estimates an equivalent dipole located at the center of mass of the actual region, and therefore produces an artifact
(3) The smoothness of αg(t) has been imposed by use of a spline basis, that limits its possible forms of the amplitude of the activity to belong to a Sobolev functional Space W2(L2), (second derivative integrable Lebesgue).
This type of model has been used solely to describe the AEP or AEKG.
This model for generators has been applied using spherical and realistic models of the Volume Conductor. However, the LF methodology has not been used, which necessitates the use of numerical algorithms for the realistic case that are not very practical since they require the evaluation of a step of the BEM in each iteration.
A TPEC was proposed for the first time in the U.S. Pat. No. 5,307,807 of Valdés-Sosa et al., providing a new modality of medical image. This invention consists on a method and system to create three-dimensional maps of the amplitude, orientation and connectivity of the generators of neural/cardiac PEC with the following innovations:
(1) An estimate of j(t) on a three-dimensional lattice inside the body is carried out.
(2) DPs of the PEC of the CRE is considered.
(3) The information available from other types of Structural Medical Images is used to specify the conductive properties of the subject's body. It is also used to define the spatial extent of a lattice for which PEC are defined (thus limiting the possible sites where the PEC are to be estimated) and to determine the orientation of the PEC. The subject's corporal geometric characteristics are incorporated in the TPEC in a quantitative manner with the use of parametric models.
(4) The use of information on the metabolism of the subject under study, available from other types of functional medical images, in order to limit even further the lattice on which j(t) is to be estimated.
(5) The generators that are modeled include not only discrete sources (dipoles) but also diffuse generators that desribe background generator noise, in occasions of higher amplitude than the discrete generators.
(6) Frequency domain estimation is introduced for the generators of CRE, in particular the cross-spectral matrix, Sj(ω), of the generators is introduced.
The improvements for the evaluation of brain activity introduced in the Valdés-Sosa et al. '807 patent were later verified by studies on the origin of abnormal electric activities in patient with focal neurological lesions. See for example Harmony, T.; Fernández-Bouzas, A.; Marosi, E.; Fernández, T.; Valdés, P.; Bosch, J.; Riera, J.; Rodriguez, M.; Reyes, A.; Silva, J; Alonso, M. and Sánchez, J. M. (1995): Frequency Source Analysis in Patients with Brain Lesions. Brain Topography, vol 8. No.2.
However, the clinical works mentioned previously and others (Valdés, P.; Carballo, J. A.; Alvarez, A.; Diaz, G. F.; Biscay, R.; Pérez, M. C.; Szava, S.; Virués, T. and Quesada, M. E.: qEEG in to public Health System. Brain Topography, Vol. 4, Nr. 4, 1992b, pp. 259–266) showed that the inventions described in the patents U.S. Pat. No. 5,282,474 and U.S. Pat. No. 5,307,807 did not take into consideration the following aspects, necessary for a more complete methodological elaboration of the TPEC:
(1) The statistical procedures described in the U.S. Pat. No. 5,282,474 were not extended for use in the TPEC. In particular, the sample space of the TPEC is a stochastic field defined on a 4 dimensions space time manifold, thus requiring for its analysis specific methods which were not taken into consideration in said patent.
(2) The regression methods used in the U.S. Pat. No. 5,282,474 to eliminate the effect of experimental concomitant variables are of a parametric global polynomial type. These are not sufficiently flexible for the description of the variations at multiple scales typical for electrophysiologicalal data.
(3) In the case of the Valdés-Sosa et al. '807 patent, the only regularization resource consists of the control of the number of dipoles whose parameters are considered. This is an implicit use of the functional RJ(j(t)|Θ) that guarantees that Ng<Θo, which is not very flexible and requires excessive human intervention.
(4) The solution for the direct problem that relates the activity of the generators with the observable signals is achieved with anatomical deconvolution. This method does not adapt to general configurations of generators and is only an inexact numeric approximation to the optimal solution.
(5) The parametric description employed to characterize the geometry of the subject's body is based on the use of quantitative descriptors of a functional form that are not sufficiently flexible to adequately describe inter-individual morphometric variability. In particular this method does not allow the inclusion of probabilistic anatomical information in those cases for which the acquisition of structural images of the particular subject under study is either undesirable or impracticable.
(6) The metabolic information available from other image modalities is only integrated as a restriction to the regions in which active dipoles will be allowed. Thus this information is not used in a statistically optimal estimation procedure.
Since the publication of the two aforementioned patents, a series of works have appeared, that while not totally solving the problems outlined above, have contributed to the elaboration of the solutions that are claimed in the present invention. The relevant developments are enumerated next, accompanied by an evaluation of their advantages and inadequacies.
The DPs used in the construction of all the TPEC until the moment are either the raw data or sufficient statistics, derived from the data, almost always some of the lower order statistical moments in the time or frequency domain. Recent works (Hernández, J, L., Valdés, P. A., and Vila, P. (1996) EEG spike and wave modeled by to stochastic limit cycle. NeuroReport, 7: 2246–2250) demonstrate the need for flexible non-parametric descriptions for the temporal sequence of the EEG/MEG/EKG/MKG that allow the description of the dynamic activity, essentially nonlinear, that is characteristic of masses of neural and heart tissue.
The formula RJ(j(t)|Θ)=∥j(t)∥ΣEEpJ includes classical regularization, which allows models of j(t) in which the activated areas can be extensive and with smooth contours. In particular Linear Distributed Solutions are limited to estimate j(t) for each time t separately. They assume that R(t)˜NRk,po(0,ΣEE) and that j(t)˜NRk,ps(0,ΣJ) with pO and pJ=2. In consequence, the estimator of generic TPEC is formulated as a Spline type solution (Riera, J. J., Aubert, E., Valdés, P., Casanova, R. and Lins, O. Discrete Spline Electric-Magnetic Tomography (DSPET) based on Realistic Neuronatomy. To appear in the proceedings of The Tenth International Conference on Biomagnetism, BIOMAG '96, Santa Fe, N. Mex., February 1996) where the smoothness of the solution is specified by the matrix ΣJJ. The estimator of the PEC has, in this case, an explicit expression that is:ĵ(t)=ΣJJ·Kt·(K·ΣJJ·Kt+ΣEE)−1·v(t)  (8)
The most significant representatives in this family of TPEC are characterized by assuming that ΣEE=σ2EEI, where σ2EE is a variance common to the sensors of the same type. The different linear solutions are characterized by the Σj postulated as indicated below.
Minimum Norm TPEC (MN) for which ΣJ=σJ2·I3·Ng. This solution requires that the configuration of generators has minimum energy (Wang J. Z., Williamson S. J. and Kaufman L. Magnetic source images determined by lead-field analysis: the unique minimum-norm least squares estimation. IEEE Trans. Biomed. Eng., 39, 7, 665–667, 1992) compatible with the likelihood. This method produces spatially widespread estimates that do not localize discrete sources correctly. Variants of this methodology were presented in the U.S. Pat. No. 5,228,443.
Weighed Minimum Norm TPEC (WMN), ΣJ=σJ2·W−2I3, where the matrix W=diag(wi)
      w    i    =                    ∑                  d          =          1                          N          d                    ⁢                                  K                                                •            i                    ⁢          d                2            W is diagonal and it contains weights. These counteract a bias that distributed solutions have of concentrating solutions near the sensors (George, J. S, P. S Lewis, H. A Schlitt, L. Kaplan, I. and C. C Wood (1993) Strategies for source space limitation in tomographic inverse procedures. In: Proc. 9th Int. Conf. On Biomagnetism, Vienna). It has been demonstrated that this method avoids spurious superficial solutions but it does not localize discrete sources with more precision.
Low resolution electric Tomography (LORETA) where:ΣJ=σJ2·(Λm·L2·Λm)−1I3 Λm=W, L3=LI3, L being the matrix of the discretized Laplacian. This solution imposes the maximum spatial smoothness compatible with the likelihood term (Pascual-Marqui, R. D., 1994. Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain, Int. J. Psychophysiol. 18:49–65. LORETA). This method localizes a single point source accurately in any part of the brain or heart. LORETA has been extended for the estimation of the cross-spectrum of the generators of the EEG. (see Casanova, R., Valdés P., Garcia F. M., Aubert E., Riera, J. J., Korin W. and Lins O. Frequency domain distributed inverse solutions. To appear in the proceedings of The Tenth International Conference on Biomagnetism, BIOMAG '96, Santa Fe, N. Mex., February 1996). This is based on the following estimator:SJ1/2(ω)=ΣJ·Kt·(K·ΣJ·Kt+ΣE)−1·Sv1/2  (9)
However, SJ1/2(ω)=ΣJ·Kt·(K·ΣJ·Kt+ΣE)−1·Sv1/2 does not correspond to a complete Bayesian model, since an a priori probability is specified for j(t), but not for Σj.
In some cases, additional restrictions can be imposed to obtain the well-known Backus and Gilbert solution, as described in U.S. Pat. No. 4,977,896.
All these Spline methods specify the membership of j(t) to a Sobolev Space Hq(L2), for integer q, which is too restrictive since the estimation of dipolar types of PEC are not allowed. In fact, the Raleigh limit is valid which states the impossibility of discriminating nearby point sources. This limit is the cause, additionally, of the appearance of “ghost” solutions that are interference artifacts due to the limited resolution of linear methods.
In search of less widespread solutions, several authors have proposed non-linear estimators of j(t). Among these (1) Matsuura and Okabe (Matsuura K. and Okabe Y. (1995) Selective Minimum-Norm Solution of the Biomagnetic Inverse Problem. IEEE Trans. BME Vol. 43 not. 6 pp. 608–615) have varied of po and pj in the Generalized gaussian distributions that define the inverse problem with the consequence that the inverse solution must now be estimated non-linearly; (2) Gorodnitsky and Bhaskar (Gorodnitsky, I. and Bhaskar D. R. 1997. Sparse signal reconstruction from limited data using FOCUSS: a reweighted minimum norm algorithm. IEEE Trans. Signal Proc. Vol. 45 (3) pp 600–616) have proposed to apply iteratively the estimator of RO+λ·RJ(j(t)|Θ) weighting each lattice point proportionally to the j(t) value estimated in the previous step. Both proposals converge algorithmically to a number of discrete generators equal to the number of sensors, therefore being in effect a more elaborate form of dipole fit. In consequence, they share with these dipolar models the defect that they can not estimate spatially distributed sources.
Philips et al. (Philips J W, Leahy R M and Mosher J C. MEG based imaging of focal neuronal current sources. (1997). IEEE trans Medical Imaging, Volume 16: 338–348) introduced for P(j(t),ΣEE|o(t))∝l(o(t)|j(t),ΣEE)·π(j(t)|Θ) the functionalRJ(j(t)|Θ)=∥ψ(t)∥Σψψpψ+V(z(t))  (10)where j(t)=ψ(t).*(z(t)13). This functional penalizes the lack of smoothness of Ψ(t) in the metric of ΣΨΨ,. It also imposes the additional requirement that there are few active Zi (which is controlled by the weights αI) which will be concentrated in neighborhoods ξI (whose dispersion is controlled by means of the parameters Q and the weights βi). This is accomplished by means of the term
      V    ⁢          (              z        ⁢                  (          t          )                    )        =            ∑              i        =        1                    N        g              ⁢          {                                    α            i                    ·                                    Z              i                        ⁢                          (              t              )                                      +                              β            i                    ·                                    [                                                ∑                                      i                    ⁢                                                                                  ∈                                                                                  ⁢                                          ξ                      i                                                                      ⁢                                                      (                                                                                            Z                          i                                                ⁢                                                  (                          t                          )                                                                    -                                                                        Z                          i                                                ⁢                                                  (                          t                          )                                                                                      )                                    2                                            ]                        Q                              }      The resulting Bayesian hierarchical model is estimated by using the method of “Mean Field Annealing.” This model, although potentially very flexible it has only been described for the case ΣΨ=σΨ2·W−2I3.
None of the methods described above models the temporal dependencies of j(t), which is equal to the use of the implicit assumption that j(t) is independent for each t. This assumption can cause considerable bias in the statistical estimates.
It should be mentioned that numerous works carry out processing of o(t) images, sometimes taking into account information contributed by other neuroimage modalities, that however do not have as an objective the estimation and ulterior use of j(t). Such is the case of the patent from Finland 925,461 and the U.S. Pat. No. 5,331,970. Since these patents do not having as an objective the obtaining of a TPEC, these types of antecedents will not be included in this analysis of the art.